Question 214015
If the sides are {{{h}}} and {{{w}}},
then {{{2h + 2w = 30}}}
{{{2w = 30 - 2h}}}
{{{w = (1/2)*(30 - 2h)}}}
{{{A = w*h}}}
{{{A = (1/2)*(30 - 2h)*h}}}
{{{A = (1/2)*(30h - 2h^2)}}}
{{{A = -h^2 + 15h}}}
The - sign tells me this is a parabola with
a peak at the top, or a maximum
The maximum occurs exactly between the
roots, or at }{{{-b/(2a)}}}
{{{b = 15}}}
{{{a = -1}}}
{{{h[max] = -b/(2a)}}} 
{{{h[max] = -15/-2}}}
{{{h[max] = 7.5}}}
and
{{{w = (1/2)*(30 - 2h)}}}
{{{w = (1/2)*(30 - 15)}}}
{{{w = 7.5}}}
A 7.5 x 7.5 rectangle, or a square, has the max area
which is{{{7.5^2 = 56.25}}} cm2
You can prove this by changing the square slightly
and keep the perimeter the same, say
7.4 x 7.6
perimeter = {{{2*7.4 + 2*7.6 = 14.8 + 15.2}}}
{{{14.8 + 15.2 = 30}}}
{{{A = 7.4*7.6}}}
{{{A = 56.24}}} (just a little smaller than 56.25)